úÎ!0¸-h:      !"#$%&'()*+,-./0123456789,Types for simple and semisimple Lie Algebras(c) Felix Springer, 2019BSD3felixspringer149@gmail.com experimentalPOSIXSafe   "A way to represent Dynkin Diagrams(c) Felix Springer, 2019BSD3felixspringer149@gmail.com experimentalPOSIXSafelie'A type that represents a Dynkin Diagram:lie¾Only reverses the first vertex in any (DynkinNode [vertex,...]). Has effect of stretching Dynkin diagrams like D3 (, which is sane). Has no purpose since variationsDynkin is more powerful?lie<Return a list of all (simpler, ) equivalent Dynkin Diagrams.;lie5Doesn't return the DynkinNode it started with (, but D does!). Doesn't find ALL variations (, but seems to work for now).  Useful functions(c) Felix Springer, 2019BSD3felixspringer149@gmail.com experimentalPOSIXSafeú !"" !QTranslation of a mathematical definition of a Lie Algebra into a Class LieAlgebra(c) Felix Springer, 2019BSD3felixspringer149@gmail.com experimentalPOSIXSafe@AC$† #lieVectorspace with Lie Bracket$lie1Addition should satisfy the following attributes.Associativity:"(x |+| y) |+| z == x |+| (y |+| z)Commutativity:x |+| y == y |+| xNeutral element 0 exists: x |+| 0 == xInverse element (-x) exists:x |+| (-x) == 0%lie>Scalar Multiplication should satisfy the following attributes.Distributivity:*a |*| (x |+| y) == (a |*| x) |+| (a |*| y)Neutral element 1 exists: 1 |*| x == x&lie4Lie Bracket should satisfy the following attributes. Bilinearity:"a |*| (x |.| y) == (a |*| x) |.| y*(x |+| y) |.| z == (x |.| z) |+| (y |.| z) Antisymmetry:x |.| y == - (y |.| x)Jacobi-Identity:7x |.| (y |.| z) + y |.| (z |.| x) + z |.| (x |.| y) = 0'lie)Ordered Basis Vectors of the Lie Algebra:linearly independentspan the whole Vectorspace(liemLinear Combination of basis vectors, where the order refers to the Basis and the values are the coefficients)lieFNatural way one would define a dual to the elements in the Lie Algebra*lieLCalculates the Trace of an object in the Dual Vectorspace of the Lie Algebra+lieRKilling Form, which is a scalar product on the Dual Vectorspace of the Lie Algebra #*+$%&'() #*+$%&'()Showcase(c) Felix Springer, 2019BSD3felixspringer149@gmail.com experimentalPOSIXSafe=?@AC(Ã,lie Vector in !³.lie~Crossproduct, which returns orthogonal vectors where the length is the area of the parallelogram spanned by the input vectors,-.,-.Showcase(c) Felix Springer, 2019BSD3felixspringer149@gmail.com experimentalPOSIXSafe=?@AC-<2lieMatrix in !^(3*/3)4lie?Complex Number (real and imaginary part have Real coefficients)5lieMatrix Multiplication6lie5Subtraction, which is Inverse to Vectorspace Addition2345642356<      !"#$%&'()*+,-./0123456789:;<=>?@!lie-0.1.0.0-Bt17OugAtiTYzUPViDdA7Lie.Classification Lie.Dynkin Lie.HelpersLie.LieAlgebraLieExample.SO3LieExample.SU3 SemiSimple DirectSumSimpleABCDE6E7E8F4G2 $fShowSimple $fEqSimple$fShowSemiSimple$fEqSemiSimple DynkinVertexDynkinVertexToShortDynkinVertexToLong DynkinNodeDynkinreverse variations$fShowDynkinVertex$fEqDynkinVertex$fShowDynkinNode$fEqDynkinNode $fShowDynkin $fEqDynkinranksimpletoDynkin LieAlgebra|+||*||.|basislinearCombination adjunctiontrace<|>VectorV crossProduct$fLieAlgebraVectorDouble $fShowVector $fEqVectorMatrixMCD||||-|$fLieAlgebraMatrixComplex $fShowMatrix $fEqMatrixreverseDynkinHelpervariationsDynkinHelper